Question

6.3×8+6×11+9×14+…

Answer 1

The given series is 3×8+6×11+9×14+............ n th term.

Now, the n th termof the series is,

a n =( n th  term of 3,6,9.......... )×( n th  term of 8,11,14.......... ) =( 3n )( 3n+5 ) =9 n 2 +15n

Sum of the series is,

S n = ∑ k=1 n a k =9 ∑ k=1 n k 2 +15 ∑ k=1 n k =9× n( n+1 )( 2n+1 ) 6 +15× n( n+1 ) 2 = 3n( n+1 )( 2n+1 ) 2 + 15n( n+1 ) 2

Solve further.

S n = 3n( n+1 ) 2 ( 2n+1+5 ) = 3n( n+1 ) 2 ( 2n+6 ) =3n( n+1 )( n+3 )

Thus, the sum of series 3×8+6×11+9×14+............ n th termis 3n( n+1 )( n+3 ).